On interpretation of force measurements in fluids: regular and thermal forces

TL;DR

This paper analyzes the forces acting on microscopic particles in fluids, deriving exact expressions for drift velocity, examining thermal force properties, and clarifying experimental interpretations within hydrodynamic theory.

Contribution

It provides an exact hydrodynamic expression for Brownian drift velocity and challenges the fundamental hypothesis of uncorrelated thermal forces in generalized Langevin equations.

Findings

Inertial and memory effects are significant at short times.

The usual assumption of uncorrelated thermal forces leads to super-diffusion.

Experimental 'color' of thermal noise is clarified and corrected.

Abstract

The paper is devoted to the problem of the determination of regular and thermal forces acting on microscopic and smaller objects in fluids. One of the methods how regular forces are determined is the measurement of the drift velocity of Brownian particles. We have obtained an exact expression for this velocity within the hydrodynamic theory of the Brownian motion. It is shown that the influence of the inertial and memory effects can be significant in the force determination when the experimental times are sufficiently short. In the second part of the work, within the same theory, we study the properties of the thermal force driving the particles in incompressible fluids. We show that the usual assumption for the Kubo's generalized Langevin equation (called the "fundamental hypothesis") that the thermal force at a time t and the velocity of the particle in preceding times are uncorrelated, leads to an unexpected super-diffusion of the particle. To obtain the Einstein diffusion at long times, the mentioned hypothesis must be abandoned, which however does not contradict to causality. Finally, we consider the "color" of thermal noise, recently measured experimentally [Th. Franosch et al., Nature 478, 85 (2011)], and correct the interpretation of these experiments.